Irony is the use of words with more than one level of meaning. On the surface, a sentence means one thing; deeper down, and half-hidden, there is a second meaning.
At its simplest, the hidden meaning is the opposite of the surface one. When a football player mis-kicks and misses the goal, his team-mate may shout "Good shot!" - literally meaning, of course, "Bad shot!", although the words mean precisely the opposite. Irony can be more subtle: it comes in many levels. When her actress friend asked Dorothy Parker whether her performance had been good, the latter replied, "Darling, good is not the word." The actress may have understood this as a compliment; the rest of us may suspect that the right word was 'dreadful'. This is a very two-edged irony.
Even more complex is the first line of Jane Austen's Pride and Prejudice: "It is a truth universally acknowledged, that a single man in possession of a good fortune, must be in want of a wife." These words introduce a novel about young women (with their mothers) seeking husbands. It is worth considering what precisely Jane Austen meant by the words "truth", "universally", "must" and "want" in the sentence quoted. None of them mean simply the literal meaning; it may be said that Austen was writing "tongue-in-cheek". Or, in academic terms, ironically.